Copyright (c) 1995 Association of Mizar Users
environ vocabulary VECTSP_1, RLSUB_1, FINSET_1, CARD_1, FINSEQ_1, RELAT_1, FUNCT_1, FINSEQ_4, RFINSEQ, BOOLE, RLVECT_2, FUNCT_2, RLVECT_1, SEQ_1, ARYTM_1, SUBSET_1, RLVECT_3, RLSUB_2, MATRLIN, TARSKI, VECTSP_9; notation TARSKI, XBOOLE_0, ENUMSET1, SUBSET_1, NUMBERS, XCMPLX_0, XREAL_0, NAT_1, RELAT_1, STRUCT_0, FUNCT_1, FUNCT_2, CARD_1, FINSET_1, FINSEQ_1, FINSEQ_3, FINSEQ_4, RFINSEQ, RLVECT_1, VECTSP_1, VECTSP_4, VECTSP_6, VECTSP_5, VECTSP_7, MATRLIN; constructors REAL_1, FINSEQ_3, RFINSEQ, VECTSP_6, VECTSP_5, VECTSP_7, MATRLIN, DOMAIN_1, RLVECT_2, PARTFUN1, XREAL_0, MEMBERED; clusters SUBSET_1, STRUCT_0, RELSET_1, FINSEQ_1, FINSET_1, MATRLIN, XREAL_0, FUNCT_2, VECTSP_1, NAT_1, MEMBERED, NUMBERS, ORDINAL2; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; definitions TARSKI; theorems AXIOMS, TARSKI, ENUMSET1, SUBSET_1, REAL_1, NAT_1, CARD_1, CARD_2, FUNCT_1, FUNCT_2, FINSEQ_1, FINSEQ_2, FINSEQ_3, FINSEQ_4, FINSET_1, RFINSEQ, RLVECT_1, VECTSP_1, VECTSP_4, VECTSP_5, VECTSP_6, VECTSP_7, MATRLIN, RELSET_1, XBOOLE_0, XBOOLE_1, RLVECT_2, VECTSP_2, ZFMISC_1, XCMPLX_0, XCMPLX_1; schemes NAT_1, FINSEQ_1, FUNCT_2, XBOOLE_0; begin reserve GF for Field, V for VectSp of GF, W for Subspace of V, x, y, y1, y2 for set, i, n, m for Nat; :: :: Preliminaries :: definition let S be non empty 1-sorted; cluster non empty Subset of S; existence proof consider A being non empty Subset of S; A is Subset of S; hence thesis; end; end; theorem Th1: for X being finite set st n <= Card X holds ex A being finite Subset of X st Card A = n proof let X be finite set such that A1: n <= Card X; consider p being FinSequence such that A2: rng p = X and A3: p is one-to-one by FINSEQ_4:73; A4: n <= len p by A1,A2,A3,FINSEQ_4:77; reconsider q = p|Seg n as FinSequence by FINSEQ_1:19; A5: len q = n by A4,FINSEQ_1:21; A6: q is one-to-one by A3,FUNCT_1:84; reconsider A = rng q as Subset of X by A2,FUNCT_1:76; Card(A) = n & A is finite by A5,A6,FINSEQ_4:77; hence thesis; end; :: More On Functions reserve f, g for Function; theorem Th2: for f st f is one-to-one holds x in rng f implies Card(f"{x}) = 1 proof let f be Function; assume A1: f is one-to-one; assume x in rng f; then f just_once_values x by A1,FINSEQ_4:10; then consider B being finite set such that A2: B = f"{x} & card B = 1 by FINSEQ_4:def 2; thus Card(f"{x}) = 1 by A2; end; theorem for f holds not x in rng f implies Card(f"{x}) = 0 by CARD_1:78,FUNCT_1:142; theorem Th4: for f, g st rng f = rng g & f is one-to-one & g is one-to-one holds f, g are_fiberwise_equipotent proof let f, g be Function such that A1: rng f = rng g and A2: f is one-to-one and A3: g is one-to-one; for x be set holds Card(f"{x}) = Card(g"{x}) proof let x; per cases; suppose x in rng f; then Card(f"{x}) = 1 & Card(g"{x}) = 1 by A1,A2,A3,Th2; hence thesis; suppose not x in rng f; then Card(f"{x}) = 0 & Card(g"{x}) = 0 by A1,CARD_1:78,FUNCT_1:142; hence thesis; end; hence f, g are_fiberwise_equipotent by RFINSEQ:def 1; end; Lm1: for X, x being set st x in X holds X \ {x} \/ {x} = X proof let X, x be set; assume x in X; then A1: {x} is Subset of X by SUBSET_1:63; {x} \/ (X \ {x}) = {x} \/ X by XBOOLE_1:39 .= X by A1,XBOOLE_1:12; hence thesis; end; :: More On Linear Combinations theorem Th5: for L being Linear_Combination of V for F, G being FinSequence of the carrier of V for P being Permutation of dom F st G = F*P holds Sum(L (#) F) = Sum(L (#) G) proof let L be Linear_Combination of V; let F, G be FinSequence of the carrier of V; let P be Permutation of dom F such that A1: G = F*P; set p = L (#) F, q = L (#) G; len F = len(L (#) F) by VECTSP_6:def 8; then A2: dom F = dom(L (#) F) by FINSEQ_3:31; then reconsider r = (L (#) F)*P as FinSequence of the carrier of V by FINSEQ_2:51; A3: len p = len F & len q = len G by VECTSP_6:def 8; A4: len G = len F by A1,FINSEQ_2:48; then A5: dom p = dom q by A3,FINSEQ_3:31; A6: dom F = dom G by A4,FINSEQ_3:31; A7: dom F = dom p by A3,FINSEQ_3:31; len r = len(L (#) F) by A2,FINSEQ_2:48; then A8: dom r = dom(L (#) F) & len r = len F by FINSEQ_3:31,VECTSP_6:def 8; now let k be Nat; assume A9: k in dom q; set l = P.k; dom P = dom F & rng P = dom F by FUNCT_2:67,def 3; then A10: l in dom F by A5,A7,A9,FUNCT_1:def 5; then reconsider l as Nat; G/.k = G.k by A5,A6,A7,A9,FINSEQ_4:def 4 .= F.(P.k) by A1,A5,A6,A7,A9,FUNCT_1:22 .= F/.l by A10,FINSEQ_4:def 4; then q.k = L.(F/.l) * (F/.l) by A9,VECTSP_6:def 8 .= (L (#) F).(P.k) by A7,A10,VECTSP_6:def 8 .= r.k by A5,A8,A9,FUNCT_1:22; hence q.k = r.k; end; then q = r by A5,A8,FINSEQ_1:17; hence Sum(p) = Sum(q) by A2,RLVECT_2:9; end; theorem Th6: for L being Linear_Combination of V for F being FinSequence of the carrier of V st Carrier(L) misses rng F holds Sum(L (#) F) = 0.V proof let L be Linear_Combination of V; let F be FinSequence of the carrier of V such that A1: Carrier(L) misses rng F; defpred P[FinSequence] means for G being FinSequence of the carrier of V st G = $1 holds Carrier(L) misses rng G implies Sum(L (#) G) = 0.V; A2: P[{}] proof let G be FinSequence of the carrier of V such that A3: G = {}; assume Carrier(L) misses rng G; L (#) G = <*>(the carrier of V) by A3,VECTSP_6:33; hence Sum(L (#) G) = 0.V by RLVECT_1:60; end; A4: for p being FinSequence, x st P[p] holds P[p^<*x*>] proof let p be FinSequence, x such that A5: P[p]; let G be FinSequence of the carrier of V; assume A6: G = p^<*x*>; then reconsider p, x'= <*x*> as FinSequence of the carrier of V by FINSEQ_1:50; x in {x} by TARSKI:def 1; then x in rng x' & rng x' c= the carrier of V by FINSEQ_1:55,def 4; then reconsider x as Vector of V; assume Carrier(L) misses rng G; then A7: {} = Carrier(L) /\ rng G by XBOOLE_0:def 7 .= Carrier(L) /\ (rng p \/ rng<*x*>) by A6,FINSEQ_1:44 .= Carrier(L) /\ (rng p \/ {x}) by FINSEQ_1:55 .= Carrier(L) /\ rng p \/ Carrier(L) /\ {x} by XBOOLE_1:23; then Carrier(L) /\ rng p = {} & Carrier(L) /\ {x} = {} by XBOOLE_1:15; then Carrier(L) misses rng p by XBOOLE_0:def 7; then A8: Sum(L (#) p) = 0.V by A5; now assume x in Carrier(L); then x in Carrier(L) & x in {x} by TARSKI:def 1; then x in Carrier(L) /\ {x} by XBOOLE_0:def 3; hence contradiction by A7,XBOOLE_1:15; end; then A9: L.x = 0.GF by VECTSP_6:20; Sum(L (#) G) = Sum((L (#) p) ^ (L (#) x')) by A6,VECTSP_6:37 .= Sum(L (#) p) + Sum(L (#) x') by RLVECT_1:58 .= 0.V + Sum(<* L.x * x *>) by A8,VECTSP_6:34 .= Sum(<* L.x * x *>) by RLVECT_1:10 .= 0.GF * x by A9,RLVECT_1:61 .= 0.V by VECTSP_1:60; hence Sum(L (#) G) = 0.V; end; for p being FinSequence holds P[p] from IndSeq(A2, A4); hence Sum(L (#) F) = 0.V by A1; end; theorem Th7: for F being FinSequence of the carrier of V st F is one-to-one for L being Linear_Combination of V st Carrier(L) c= rng F holds Sum(L (#) F) = Sum(L) proof let F be FinSequence of the carrier of V such that A1: F is one-to-one; let L be Linear_Combination of V such that A2: Carrier(L) c= rng F; consider G being FinSequence of the carrier of V such that A3: G is one-to-one and A4: rng G = Carrier(L) and A5: Sum(L) = Sum(L (#) G) by VECTSP_6:def 9; reconsider A = rng G as Subset of rng F by A2,A4; consider P being Permutation of dom F such that A6: (F - A`) ^ (F - A) = F*P by A1,MATRLIN:8; set F1 = F - A`; rng F c= rng F; then reconsider X = rng F as Subset of rng F; X \ A` = X /\ A`` by SUBSET_1:32 .= A by XBOOLE_1:28; then rng F1 = rng G & F1 is one-to-one by A1,FINSEQ_3:72,94; then F1, G are_fiberwise_equipotent by A3,Th4; then consider Q being Permutation of dom G such that A7: F1 = G*Q by RFINSEQ:17; reconsider F1, F2 = F - A as FinSequence of the carrier of V by FINSEQ_3:93; rng F2 = rng F \ rng G by FINSEQ_3:72; then A8: rng F2 misses rng G by XBOOLE_1:79; Sum(L (#) F) = Sum(L (#) (F1^F2)) by A6,Th5 .= Sum((L (#) F1) ^ (L (#) F2)) by VECTSP_6:37 .= Sum(L (#) F1) + Sum(L (#) F2) by RLVECT_1:58 .= Sum(L (#) F1) + 0.V by A4,A8,Th6 .= Sum(L (#) G) + 0.V by A7,Th5 .= Sum(L) by A5,RLVECT_1:10; hence thesis; end; theorem Th8: for L being Linear_Combination of V for F being FinSequence of the carrier of V holds ex K being Linear_Combination of V st Carrier(K) = rng F /\ Carrier(L) & L (#) F = K (#) F proof let L be Linear_Combination of V, F be FinSequence of the carrier of V; defpred P[set, set] means $1 is Vector of V implies ($1 in rng F & $2 = L.$1) or (not $1 in rng F & $2 = 0.GF); A1: for x st x in the carrier of V ex y st y in the carrier of GF & P[x, y] proof let x; assume x in the carrier of V; then reconsider x'= x as Vector of V; per cases; suppose x in rng F; then P[x, L.x']; hence thesis; suppose not x in rng F; hence thesis; end; ex K being Function of the carrier of V, the carrier of GF st for x st x in the carrier of V holds P[x, K.x] from FuncEx1(A1); then consider K being Function of the carrier of V, the carrier of GF such that A2: for x st x in the carrier of V holds P[x, K.x]; rng F is Subset of V by FINSEQ_1:def 4; then reconsider R = rng F as finite Subset of V; A3: now let v be Vector of V; assume A4: not v in R /\ Carrier(L); per cases by A4,XBOOLE_0:def 3; suppose not v in R; hence K.v = 0.GF by A2; suppose A5: not v in Carrier(L); (P[v, K.v] & P[v, L.v]) or (P[v, K.v] & P[v, 0.GF]) by A2; hence K.v = 0.GF by A5,VECTSP_6:20; end; reconsider K as Element of Funcs(the carrier of V, the carrier of GF) by FUNCT_2:11; reconsider K as Linear_Combination of V by A3,VECTSP_6:def 4; take K; A6: Carrier(K) = rng F /\ Carrier(L) proof now let v be set; assume v in Carrier(K); then consider v' being Vector of V such that A7: v'= v & K.v' <> 0.GF by VECTSP_6:19; thus v in rng F /\ Carrier(L) by A3,A7; end; then A8: Carrier(K) c= rng F /\ Carrier(L) by TARSKI:def 3; now let v be set; assume v in rng F /\ Carrier(L); then A9: v in R & v in Carrier(L) by XBOOLE_0:def 3; then reconsider v'= v as Vector of V; K.v' = L.v' & L.v' <> 0.GF by A2,A9,VECTSP_6:20; hence v in Carrier(K) by VECTSP_6:19; end; then rng F /\ Carrier(L) c= Carrier(K) by TARSKI:def 3; hence thesis by A8,XBOOLE_0:def 10; end; L (#) F = K (#) F proof set p = L (#) F, q = K (#) F; len p = len F & len q = len F by VECTSP_6:def 8; then A10: dom p = dom q & dom p = dom F by FINSEQ_3:31; now let k be Nat; assume A11: k in dom p; set u = F/.k; F/.k = F.k by A10,A11,FINSEQ_4:def 4; then A12: P[u, K.u] & P[u, L.u] by A2,A10,A11,FUNCT_1:def 5; p.k = L.u * u & q.k = K.u * u by A10,A11,VECTSP_6:def 8; hence p.k = q.k by A12; end; hence thesis by A10,FINSEQ_1:17; end; hence thesis by A6; end; theorem Th9: for L being Linear_Combination of V for A being Subset of V for F being FinSequence of the carrier of V st rng F c= the carrier of Lin(A) holds ex K being Linear_Combination of A st Sum(L (#) F) = Sum(K) proof let L be Linear_Combination of V; let A be Subset of V; defpred P[Nat] means for F being FinSequence of the carrier of V st rng F c= the carrier of Lin(A) & len F = $1 holds ex K being Linear_Combination of A st Sum(L (#) F) = Sum(K); A1: P[0] proof let F be FinSequence of the carrier of V; assume rng F c= the carrier of Lin(A) & len F = 0; then F = <*>(the carrier of V) by FINSEQ_1:25; then L (#) F = <*>(the carrier of V) by VECTSP_6:33; then A2: Sum(L (#) F) = 0.V by RLVECT_1:60 .= Sum(ZeroLC(V)) by VECTSP_6:41; ZeroLC(V) is Linear_Combination of A by VECTSP_6:26; hence thesis by A2; end; A3: for n being Nat st P[n] holds P[n + 1] proof let n be Nat; assume A4: P[n]; let F be FinSequence of the carrier of V such that A5: rng F c= the carrier of Lin(A) and A6: len F = n + 1; consider G being FinSequence, x being set such that A7: F = G^<*x*> by A6,FINSEQ_2:21; reconsider G, x'= <*x*> as FinSequence of the carrier of V by A7,FINSEQ_1:50; rng(G^<*x*>) = rng G \/ rng <*x*> by FINSEQ_1:44 .= rng G \/ {x} by FINSEQ_1:55; then rng G c= rng F & {x} c= rng F by A7,XBOOLE_1:7; then A8: rng G c= the carrier of Lin(A) & {x} c= the carrier of Lin(A) by A5,XBOOLE_1:1; then x in {x} implies x in the carrier of Lin(A); then A9: x in Lin(A) by RLVECT_1:def 1,TARSKI:def 1; then consider LA1 being Linear_Combination of A such that A10: x = Sum(LA1) by VECTSP_7:12; x in V by A9,VECTSP_4:17; then reconsider x as Vector of V by RLVECT_1:def 1; len(G^<*x*>) = len G + len <*x*> by FINSEQ_1:35 .= len G + 1 by FINSEQ_1:56; then len G = n by A6,A7,XCMPLX_1:2; then consider LA2 being Linear_Combination of A such that A11: Sum(L (#) G) = Sum(LA2) by A4,A8; A12: Sum(L (#) F) = Sum((L (#) G) ^ (L (#) x')) by A7,VECTSP_6:37 .= Sum(LA2) + Sum(L (#) x') by A11,RLVECT_1:58 .= Sum(LA2) + Sum(<*L.x * x*>) by VECTSP_6:34 .= Sum(LA2) + L.x * Sum(LA1) by A10,RLVECT_1:61 .= Sum(LA2) + Sum(L.x * LA1) by VECTSP_6:78 .= Sum(LA2 + L.x * LA1) by VECTSP_6:77; L.x * LA1 is Linear_Combination of A by VECTSP_6:61; then LA2 + L.x * LA1 is Linear_Combination of A by VECTSP_6:52; hence thesis by A12; end; A13: for n being Nat holds P[n] from Ind(A1, A3); let F be FinSequence of the carrier of V; assume A14: rng F c= the carrier of Lin(A); len F is Nat; hence thesis by A13,A14; end; theorem Th10: for L being Linear_Combination of V for A being Subset of V st Carrier(L) c= the carrier of Lin(A) holds ex K being Linear_Combination of A st Sum(L) = Sum(K) proof let L be Linear_Combination of V, A be Subset of V; consider F being FinSequence of the carrier of V such that F is one-to-one and A1: rng F = Carrier(L) and A2: Sum(L) = Sum(L (#) F) by VECTSP_6:def 9; assume Carrier(L) c= the carrier of Lin(A); then consider K being Linear_Combination of A such that A3: Sum(L (#) F) = Sum(K) by A1,Th9; take K; thus thesis by A2,A3; end; theorem Th11: for L being Linear_Combination of V st Carrier(L) c= the carrier of W for K being Linear_Combination of W st K = L|the carrier of W holds Carrier(L) = Carrier(K) & Sum(L) = Sum(K) proof let L be Linear_Combination of V such that A1: Carrier(L) c= the carrier of W; let K be Linear_Combination of W such that A2: K = L|the carrier of W; A3: the carrier of W c= the carrier of V by VECTSP_4:def 2; A4: dom K = the carrier of W by FUNCT_2:def 1; now let x be set; assume x in Carrier(K); then consider w being Vector of W such that A5: x = w and A6: K.w <> 0.GF by VECTSP_6:19; L.w <> 0.GF & w is Vector of V by A2,A4,A6,FUNCT_1:70,VECTSP_4:18; hence x in Carrier(L) by A5,VECTSP_6:19; end; then A7: Carrier(K) c= Carrier(L) by TARSKI:def 3; now let x be set; assume A8: x in Carrier(L); then consider v being Vector of V such that A9: x = v and A10: L.v <> 0.GF by VECTSP_6:19; K.v <> 0.GF by A1,A2,A4,A8,A9,A10,FUNCT_1:70; hence x in Carrier(K) by A1,A8,A9,VECTSP_6:19; end; then A11: Carrier(L) c= Carrier(K) by TARSKI:def 3; then A12: Carrier(K) = Carrier(L) by A7,XBOOLE_0:def 10; consider F being FinSequence of the carrier of V such that A13: F is one-to-one and A14: rng F = Carrier(L) and A15: Sum(L) = Sum(L (#) F) by VECTSP_6:def 9; consider G being FinSequence of the carrier of W such that A16: G is one-to-one and A17: rng G = Carrier(K) and A18: Sum(K) = Sum(K (#) G) by VECTSP_6:def 9; F, G are_fiberwise_equipotent by A12,A13,A14,A16,A17,Th4; then consider P being Permutation of dom G such that A19: F = G*P by RFINSEQ:17; set p = L (#) F; len G = len F by A19,FINSEQ_2:48; then dom G = dom F & len G = len (L (#) F) by FINSEQ_3:31,VECTSP_6:def 8; then A20: dom p = dom G & dom G = dom F by FINSEQ_3:31; len(K (#) G) = len G by VECTSP_6:def 8; then A21: dom(K (#) G) = dom G by FINSEQ_3:31; then reconsider q = (K (#) G)*P as FinSequence of the carrier of W by FINSEQ_2:51; len q = len (K (#) G) by A21,FINSEQ_2:48; then dom q = dom(K (#) G) & len q = len G by FINSEQ_3:31,VECTSP_6:def 8; then A22: dom q = dom(K (#) G) & dom q = dom G by FINSEQ_3:31; now let i; assume A23: i in dom p; set v = F/.i; A24: F/.i = F.i by A20,A23,FINSEQ_4:def 4; set j = P.i; dom P = dom G & rng P = dom G by FUNCT_2:67,def 3; then A25: j in dom G by A20,A23,FUNCT_1:def 5; then reconsider j as Nat; A26: G/.j = G.(P.i) by A25,FINSEQ_4:def 4 .= v by A19,A20,A23,A24,FUNCT_1:22; v in rng F by A20,A23,A24,FUNCT_1:def 5; then reconsider w = v as Vector of W by A12,A14; q.i = (K (#) G).j by A20,A22,A23,FUNCT_1:22 .= K.w * w by A22,A25,A26,VECTSP_6:def 8 .= L.v * w by A2,A4,FUNCT_1:70 .= L.v * v by VECTSP_4:22; hence p.i = q.i by A23,VECTSP_6:def 8; end; then A27: L (#) F = (K (#) G)*P by A20,A22,FINSEQ_1:17; len G = len (K (#) G) by VECTSP_6:def 8; then dom G = dom (K (#) G) by FINSEQ_3:31; then reconsider P as Permutation of dom (K (#) G); q = (K (#) G)*P; then A28: Sum(K (#) G) = Sum(q) by RLVECT_2:9; rng q c= the carrier of W by FINSEQ_1:def 4; then rng q c= the carrier of V by A3,XBOOLE_1:1; then reconsider q'= q as FinSequence of the carrier of V by FINSEQ_1:def 4; consider f being Function of NAT, the carrier of W such that A29: Sum(q) = f.(len q) and A30: f.0 = 0.W and A31: for i being Nat, w being Vector of W st i < len q & w = q.(i + 1) holds f.(i + 1) = f.i + w by RLVECT_1:def 12; dom f = NAT & rng f c= the carrier of W by FUNCT_2:def 1,RELSET_1:12; then dom f = NAT & rng f c= the carrier of V by A3,XBOOLE_1:1; then reconsider f'= f as Function of NAT, the carrier of V by FUNCT_2:4; A32: f'.0 = 0.V by A30,VECTSP_4:19; A33: for i being Nat, v being Vector of V st i < len q' & v = q'.(i + 1) holds f'.(i + 1) = f'.i + v proof let i be Nat, v be Vector of V; assume A34: i < len q' & v = q'.(i + 1); then 1 <= i + 1 & i + 1 <= len q by NAT_1:29,38; then i + 1 in dom q by FINSEQ_3:27; then reconsider v'= v as Vector of W by A34,FINSEQ_2:13; f.(i + 1) = f.i + v' by A31,A34; hence f'.(i + 1) = f'.i + v by VECTSP_4:21; end; f'.len q' is Element of V; hence thesis by A7,A11,A15,A18,A27,A28,A29,A32,A33,RLVECT_1:def 12,XBOOLE_0: def 10; end; theorem Th12: for K being Linear_Combination of W holds ex L being Linear_Combination of V st Carrier(K) = Carrier(L) & Sum(K) = Sum (L) proof let K be Linear_Combination of W; defpred P[set, set] means ($1 in W & $2 = K.$1) or (not $1 in W & $2 = 0.GF); A1: for x st x in the carrier of V ex y st y in the carrier of GF & P[x, y] proof let x; assume x in the carrier of V; then reconsider x as Vector of V; per cases; suppose A2: x in W; then reconsider x as Vector of W by RLVECT_1:def 1; P[x, K.x] by A2; hence thesis; suppose not x in W; hence thesis; end; ex L being Function of the carrier of V, the carrier of GF st for x st x in the carrier of V holds P[x, L.x] from FuncEx1(A1); then consider L being Function of the carrier of V, the carrier of GF such that A3: for x st x in the carrier of V holds P[x, L.x]; A4: the carrier of W c= the carrier of V by VECTSP_4:def 2; then Carrier(K) c= the carrier of V by XBOOLE_1:1; then reconsider C = Carrier(K) as finite Subset of V; A5: now let v be Vector of V; assume not v in C; then (P[v, K.v] & not v in C & v in the carrier of W) or P[v, 0.GF] by RLVECT_1:def 1; then (P[v, K.v] & K.v = 0.GF) or P[v, 0.GF] by VECTSP_6:20; hence L.v = 0.GF by A3; end; L is Element of Funcs(the carrier of V, the carrier of GF) by FUNCT_2:11; then reconsider L as Linear_Combination of V by A5,VECTSP_6:def 4; take L; now let x be set; assume A6: x in Carrier(L) & not x in the carrier of W; then consider v being Vector of V such that A7: x = v & L.v <> 0.GF by VECTSP_6:19; P[v, 0.GF] & P[v, L.v] by A3,A6,A7,RLVECT_1:def 1; hence contradiction by A7; end; then A8: Carrier(L) c= the carrier of W by TARSKI:def 3; reconsider K'= K as Function of the carrier of W, the carrier of GF; reconsider L'= L|the carrier of W as Function of the carrier of W, the carrier of GF by A4,FUNCT_2:38; now let x be set; assume A9: x in the carrier of W; then P[x, K.x] & P[x, L.x] by A3,A4,RLVECT_1:def 1; hence K'.x = L'.x by A9,FUNCT_1:72; end; then K' = L' by FUNCT_2:18; hence thesis by A8,Th11; end; theorem Th13: for L being Linear_Combination of V st Carrier(L) c= the carrier of W holds ex K being Linear_Combination of W st Carrier(K) = Carrier(L) & Sum(K) = Sum (L) proof let L be Linear_Combination of V; assume A1: Carrier(L) c= the carrier of W; then reconsider C = Carrier(L) as finite Subset of W; the carrier of W c= the carrier of V by VECTSP_4:def 2; then reconsider K = L|the carrier of W as Function of the carrier of W, the carrier of GF by FUNCT_2:38; A2: dom K = the carrier of W by FUNCT_2:def 1; A3: now let w be Vector of W; assume not w in C; then not w in C & w is Vector of V by VECTSP_4:18; then L.w = 0.GF by VECTSP_6:20; hence K.w = 0.GF by A2,FUNCT_1:70; end; K is Element of Funcs(the carrier of W, the carrier of GF) by FUNCT_2:11; then reconsider K as Linear_Combination of W by A3,VECTSP_6:def 4; take K; thus thesis by A1,Th11; end; :: More On Linear Independance & Basis theorem Th14: for I being Basis of V, v being Vector of V holds v in Lin(I) proof let I be Basis of V, v be Vector of V; v in the VectSpStr of V by RLVECT_1:def 1; hence v in Lin(I) by VECTSP_7:def 3; end; theorem Th15: for A being Subset of W st A is linearly-independent holds ex B being Subset of V st B is linearly-independent & B = A proof let A be Subset of W; assume A1: A is linearly-independent; the carrier of W c= the carrier of V by VECTSP_4:def 2; then A c= the carrier of V by XBOOLE_1:1; then reconsider A'= A as Subset of V; take A'; now assume A' is linearly-dependent; then consider L being Linear_Combination of A' such that A2: Sum(L) = 0.V & Carrier(L) <> {} by VECTSP_7:def 1; A3: Carrier(L) c= A' & A' c= A by VECTSP_6:def 7; then Carrier(L) c= the carrier of W by XBOOLE_1:1; then consider LW being Linear_Combination of W such that A4: Carrier(LW) = Carrier(L) & Sum(LW) = Sum(L) by Th13; reconsider LW as Linear_Combination of A by A3,A4,VECTSP_6:def 7; Sum(LW) = 0.W & Carrier(LW) <> {} by A2,A4,VECTSP_4:19; hence contradiction by A1,VECTSP_7:def 1; end; hence thesis; end; theorem Th16: for A being Subset of V st A is linearly-independent & A c= the carrier of W holds ex B being Subset of W st B is linearly-independent & B = A proof let A be Subset of V such that A1: A is linearly-independent and A2: A c= the carrier of W; reconsider A'= A as Subset of W by A2; take A'; now assume A' is linearly-dependent; then consider K being Linear_Combination of A' such that A3: Sum(K) = 0.W & Carrier(K) <> {} by VECTSP_7:def 1; consider L being Linear_Combination of V such that A4: Carrier(L) = Carrier(K) & Sum(L) = Sum(K) by Th12; Carrier(L) c= A by A4,VECTSP_6:def 7; then reconsider L as Linear_Combination of A by VECTSP_6:def 7; Sum(L) = 0.V & Carrier(L) <> {} by A3,A4,VECTSP_4:19; hence contradiction by A1,VECTSP_7:def 1; end; hence thesis; end; theorem Th17: for A being Basis of W ex B being Basis of V st A c= B proof let A be Basis of W; A is Subset of W & A is linearly-independent by VECTSP_7:def 3; then consider B being Subset of V such that A1: B is linearly-independent and A2: B = A by Th15; consider I being Basis of V such that A3: B c= I by A1,VECTSP_7:27; take I; thus thesis by A2,A3; end; theorem Th18: for A being Subset of V st A is linearly-independent for v being Vector of V st v in A for B being Subset of V st B = A \ {v} holds not v in Lin(B) proof let A be Subset of V such that A1: A is linearly-independent; let v be Vector of V such that A2: v in A; let B be Subset of V such that A3: B = A \ {v}; assume v in Lin(B); then consider L being Linear_Combination of B such that A4: v = Sum(L) by VECTSP_7:12; v in {v} by TARSKI:def 1; then v in Lin({v}) by VECTSP_7:13; then consider K being Linear_Combination of {v} such that A5: v = Sum(K) by VECTSP_7:12; A6: 0.V = Sum(L) + (- Sum(K)) by A4,A5,RLVECT_1:def 10 .= Sum(L) + Sum(-K) by VECTSP_6:79 .= Sum(L + (- K)) by VECTSP_6:77 .= Sum(L - K) by VECTSP_6:def 14; A7: {v} is Subset of A by A2,SUBSET_1:63; then A8: {v} c= A & B c= A by A3,XBOOLE_1:36; {v} is linearly-independent by A1,A7,VECTSP_7:2; then v <> 0.V by VECTSP_7:5; then Carrier(L) <> {} by A4,VECTSP_6:45; then consider w being set such that A9: w in Carrier(L) by XBOOLE_0:def 1; A10: Carrier(L - K) c= Carrier(L) \/ Carrier(K) by VECTSP_6:74; A11: Carrier(L) c= B & Carrier(K) c= {v} by VECTSP_6:def 7; then A12: Carrier(L) \/ Carrier(K) c= B \/ {v} by XBOOLE_1:13; B \/ {v} c= A \/ A by A8,XBOOLE_1:13; then Carrier(L) \/ Carrier(K) c= A by A12,XBOOLE_1:1; then Carrier(L - K) c= A by A10,XBOOLE_1:1; then A13: L - K is Linear_Combination of A by VECTSP_6:def 7; A14: for x being Vector of V holds x in Carrier(L) implies K.x = 0.GF proof let x be Vector of V; assume x in Carrier(L); then not x in Carrier(K) by A3,A11,XBOOLE_0:def 4; hence K.x = 0.GF by VECTSP_6:20; end; now let x be set such that A15: x in Carrier(L) and A16: not x in Carrier(L - K); reconsider x as Vector of V by A15; A17: L.x <> 0.GF by A15,VECTSP_6:20; (L - K).x = L.x - K.x by VECTSP_6:73 .= L.x - 0.GF by A14,A15 .= L.x + (-0.GF) by RLVECT_1:def 11 .= L.x + 0.GF by RLVECT_1:25 .= L.x by RLVECT_1:10; hence contradiction by A16,A17,VECTSP_6:20; end; then Carrier(L - K) is non empty by A9; hence contradiction by A1,A6,A13,VECTSP_7:def 1; end; theorem Th19: for I being Basis of V for A being non empty Subset of V st A misses I for B being Subset of V st B = I \/ A holds B is linearly-dependent proof let I be Basis of V; let A be non empty Subset of V such that A1: A misses I; let B be Subset of V such that A2: B = I \/ A; assume A3: B is linearly-independent; consider v being set such that A4: v in A by XBOOLE_0:def 1; reconsider v as Vector of V by A4; reconsider Bv = B \ {v} as Subset of V; A c= B by A2,XBOOLE_1:7; then A5: not v in Lin(Bv) by A3,A4,Th18; I c= B by A2,XBOOLE_1:7; then I \ {v} c= B \ {v} & not v in I by A1,A4,XBOOLE_0:3,XBOOLE_1:33; then I c= Bv by ZFMISC_1:65; then Lin(I) is Subspace of Lin(Bv) & v in Lin(I) by Th14,VECTSP_7:18; hence contradiction by A5,VECTSP_4:16; end; theorem Th20: for A being Subset of V st A c= the carrier of W holds Lin(A) is Subspace of W proof let A be Subset of V; assume A1: A c= the carrier of W; now let w be set; assume w in the carrier of Lin(A); then w in Lin(A) by RLVECT_1:def 1; then consider L being Linear_Combination of A such that A2: w = Sum(L) by VECTSP_7:12; Carrier(L) c= A by VECTSP_6:def 7; then Carrier(L) c= the carrier of W by A1,XBOOLE_1:1; then consider K being Linear_Combination of W such that A3: Carrier(K) = Carrier(L) & Sum(L) = Sum(K) by Th13; thus w in the carrier of W by A2,A3; end; then the carrier of Lin(A) c= the carrier of W by TARSKI:def 3; hence Lin(A) is Subspace of W by VECTSP_4:35; end; theorem Th21: for A being Subset of V, B being Subset of W st A = B holds Lin(A) = Lin(B) proof let A be Subset of V, B be Subset of W; assume A1: A = B; now let x be set; assume x in the carrier of Lin(A); then x in Lin(A) by RLVECT_1:def 1; then consider L being Linear_Combination of A such that A2: x = Sum(L) by VECTSP_7:12; Carrier(L) c= A & A c= the carrier of W by A1,VECTSP_6:def 7; then Carrier(L) c= the carrier of W by XBOOLE_1:1; then consider K being Linear_Combination of W such that A3: Carrier(K) = Carrier(L) & Sum(K) = Sum(L) by Th13; Carrier(K) c= B by A1,A3,VECTSP_6:def 7; then reconsider K as Linear_Combination of B by VECTSP_6:def 7; x = Sum(K) by A2,A3; then x in Lin(B) by VECTSP_7:12; hence x in the carrier of Lin(B) by RLVECT_1:def 1; end; then A4: the carrier of Lin(A) c= the carrier of Lin(B) by TARSKI:def 3; now let x be set; assume x in the carrier of Lin(B); then x in Lin(B) by RLVECT_1:def 1; then consider K being Linear_Combination of B such that A5: x = Sum(K) by VECTSP_7:12; consider L being Linear_Combination of V such that A6: Carrier(L) = Carrier(K) & Sum(L) = Sum(K) by Th12; Carrier(L) c= A by A1,A6,VECTSP_6:def 7; then reconsider L as Linear_Combination of A by VECTSP_6:def 7; x = Sum(L) by A5,A6; then x in Lin(A) by VECTSP_7:12; hence x in the carrier of Lin(A) by RLVECT_1:def 1; end; then the carrier of Lin(B) c= the carrier of Lin(A) by TARSKI:def 3; then A7: the carrier of Lin(A) = the carrier of Lin(B) by A4,XBOOLE_0:def 10; reconsider B'= Lin(B), V'= V as VectSp of GF; B' is Subspace of V' by VECTSP_4:34; hence Lin(A) = Lin(B) by A7,VECTSP_4:37; end; begin :: :: Steinitz Theorem :: :: Exchange Lemma theorem Th22: for A, B being finite Subset of V for v being Vector of V st v in Lin(A \/ B) & not v in Lin(B) holds ex w being Vector of V st w in A & w in Lin(A \/ B \ {w} \/ {v}) proof let A, B be finite Subset of V; let v be Vector of V such that A1: v in Lin(A \/ B) and A2: not v in Lin(B); consider L being Linear_Combination of (A \/ B) such that A3: v = Sum(L) by A1,VECTSP_7:12; A4: Carrier(L) c= A \/ B by VECTSP_6:def 7; now assume A5: for w being Vector of V st w in A holds L.w = 0.GF; now let x be set; assume A6: x in Carrier(L) & x in A; then ex u being Vector of V st x = u & L.u <> 0.GF by VECTSP_6:19; hence contradiction by A5,A6; end; then Carrier(L) misses A by XBOOLE_0:3; then Carrier(L) c= B by A4,XBOOLE_1:73; then L is Linear_Combination of B by VECTSP_6:def 7; hence contradiction by A2,A3,VECTSP_7:12; end; then consider w being Vector of V such that A7: w in A and A8: L.w <> 0.GF; take w; set a = L.w; consider F being FinSequence of the carrier of V such that A9: F is one-to-one and A10: rng F = Carrier(L) and A11: Sum(L) = Sum(L (#) F) by VECTSP_6:def 9; A12: w in Carrier(L) by A8,VECTSP_6:19; then A13: F = (F -| w) ^ <* w *> ^ (F |-- w) by A10,FINSEQ_4:66; reconsider Fw1 = (F -| w) as FinSequence of the carrier of V by A10,A12,FINSEQ_4:53; reconsider Fw2 = (F |-- w) as FinSequence of the carrier of V by A10,A12,FINSEQ_4:65; F = Fw1 ^ (<* w *> ^ Fw2) by A13,FINSEQ_1:45; then L (#) F = (L (#) Fw1) ^ (L (#) (<* w *> ^ Fw2)) by VECTSP_6:37 .= (L (#) Fw1) ^ ((L (#) <* w *>) ^ (L (#) Fw2)) by VECTSP_6:37 .= (L (#) Fw1) ^ (L (#) <* w *>) ^ (L (#) Fw2) by FINSEQ_1:45 .= (L (#) Fw1) ^ <* a*w *> ^ (L (#) Fw2) by VECTSP_6:34; then A14: Sum(L (#) F) = Sum((L (#) Fw1) ^ (<* a*w *> ^ (L (#) Fw2))) by FINSEQ_1:45 .= Sum(L (#) Fw1) + Sum(<* a*w *> ^ (L (#) Fw2)) by RLVECT_1:58 .= Sum(L (#) Fw1) + (Sum(<* a*w *>) + Sum(L (#) Fw2)) by RLVECT_1:58 .= Sum(L (#) Fw1) + (Sum(L (#) Fw2) + a*w) by RLVECT_1:61 .= (Sum(L (#) Fw1) + Sum(L (#) Fw2)) + a*w by VECTSP_1:7 .= Sum((L (#) Fw1) ^ (L (#) Fw2)) + a*w by RLVECT_1:58 .= Sum(L (#) (Fw1 ^ Fw2)) + a*w by VECTSP_6:37; set Fw = Fw1 ^ Fw2; consider K being Linear_Combination of V such that A15: Carrier(K) = rng Fw /\ Carrier(L) & L (#) Fw = K (#) Fw by Th8; F just_once_values w by A9,A10,A12,FINSEQ_4:10; then A16: Fw = F - {w} by FINSEQ_4:69; then A17: rng Fw = rng F \ {w} by FINSEQ_3:72; A18: rng Fw = Carrier(L) \ {w} by A10,A16,FINSEQ_3:72; rng Fw c= Carrier(L) by A10,A17,XBOOLE_1:36; then A19: Carrier(K) = rng Fw by A15,XBOOLE_1:28; A20: Carrier(L) \ {w} c= A \/ B \ {w} by A4,XBOOLE_1:33; then reconsider K as Linear_Combination of (A \/ B \ {w}) by A18,A19,VECTSP_6 :def 7; Fw1 is one-to-one & Fw2 is one-to-one & rng Fw1 misses rng Fw2 by A9,A10,A12,FINSEQ_4:67,68,72; then Fw is one-to-one by FINSEQ_3:98; then Sum(K (#) Fw) = Sum(K) by A19,VECTSP_6:def 9; then a"*v = a"*Sum(K) + a"*(a*w) by A3,A11,A14,A15,VECTSP_1:def 26 .= a"*Sum(K) + w by A8,VECTSP_1:67; then A21: w = a"*v - a"*Sum(K) by VECTSP_2:22 .= a"*(v - Sum(K)) by VECTSP_1:70 .= a"*(-Sum(K) + v) by RLVECT_1:def 11; v in {v} by TARSKI:def 1; then v in Lin({v}) by VECTSP_7:13; then consider Lv being Linear_Combination of {v} such that A22: v = Sum(Lv) by VECTSP_7:12; A23: w = a"*(Sum(-K) + Sum(Lv)) by A21,A22,VECTSP_6:79 .= a"*Sum(-K + Lv) by VECTSP_6:77 .= Sum(a"*(-K + Lv)) by VECTSP_6:78; set LC = a"*(-K + Lv); A24: Carrier (a"*(-K + Lv)) c= Carrier(-K + Lv) by VECTSP_6:58; A25: Carrier(Lv) c= {v} by VECTSP_6:def 7; Carrier (-K + Lv) c= Carrier(-K) \/ Carrier(Lv) by VECTSP_6:51; then A26: Carrier (-K + Lv) c= Carrier(K) \/ Carrier(Lv) by VECTSP_6:69; Carrier(K) \/ Carrier(Lv) c= A \/ B \ {w} \/ {v} by A18,A19,A20,A25, XBOOLE_1:13; then Carrier (-K + Lv) c= A \/ B \ {w} \/ {v} by A26,XBOOLE_1:1; then Carrier (LC) c= A \/ B \ {w} \/ {v} by A24,XBOOLE_1:1; then LC is Linear_Combination of (A \/ B \ {w} \/ {v}) by VECTSP_6:def 7; hence thesis by A7,A23,VECTSP_7:12; end; :: Steinitz Theorem theorem Th23: for A, B being finite Subset of V st the VectSpStr of V = Lin(A) & B is linearly-independent holds Card B <= Card A & ex C being finite Subset of V st C c= A & Card C = Card A - Card B & the VectSpStr of V = Lin(B \/ C) proof let A, B be finite Subset of V such that A1: the VectSpStr of V = Lin(A) and A2: B is linearly-independent; defpred P[Nat] means for n being Nat for A, B being finite Subset of V st card(A) = n & card(B) = $1 & the VectSpStr of V = Lin(A) & B is linearly-independent holds $1 <= n & ex C being finite Subset of V st C c= A & card(C) = n - $1 & the VectSpStr of V = Lin(B \/ C); A3: P[0] proof let n be Nat; let A, B be finite Subset of V such that A4: card(A) = n & card(B) = 0 & the VectSpStr of V = Lin(A) & B is linearly-independent; B = {} by A4,CARD_2:59; then A = B \/ A; hence thesis by A4,NAT_1:18; end; A5: for m being Nat st P[m] holds P[m + 1] proof let m be Nat such that A6: P[m]; let n be Nat; let A, B be finite Subset of V such that A7: card(A) = n and A8: card(B) = m + 1 and A9: the VectSpStr of V = Lin(A) and A10: B is linearly-independent; consider v being set such that A11: v in B by A8,CARD_1:47,XBOOLE_0:def 1; reconsider v as Vector of V by A11; {v} is Subset of B by A11,SUBSET_1:63; then A12: card(B \ {v}) = card(B) - card({v}) by CARD_2:63 .= m + 1 - 1 by A8,CARD_1:79 .= m by XCMPLX_1:26; set Bv = B \ {v}; A13: Bv c= B by XBOOLE_1:36; then A14: Bv is linearly-independent by A10,VECTSP_7:2; A15: not v in Lin(Bv) by A10,A11,Th18; now assume m = n; then consider C being finite Subset of V such that A16: C c= A & card(C) = m - m & the VectSpStr of V = Lin(Bv \/ C) by A6,A7,A9,A12,A14; card(C) = 0 by A16,XCMPLX_1:14; then C = {} by CARD_2:59; then Bv is Basis of V by A14,A16,VECTSP_7:def 3; hence contradiction by A15,Th14; end; then m <> n & m <= n by A6,A7,A9,A12,A14; then A17: m < n by REAL_1:def 5; consider C being finite Subset of V such that A18: C c= A and A19: card(C) = n - m and A20: the VectSpStr of V = Lin(Bv \/ C) by A6,A7,A9,A12,A14; v in Lin(Bv \/ C) by A20,RLVECT_1:def 1; then consider w being Vector of V such that A21: w in C and A22: w in Lin(C \/ Bv \ {w} \/ {v}) by A15,Th22; set Cw = C \ {w}; Cw c= C by XBOOLE_1:36; then A23: Cw c= A by A18,XBOOLE_1:1; {w} is Subset of C by A21,SUBSET_1:63; then A24: card(Cw) = card(C) - card({w}) by CARD_2:63 .= n - m - 1 by A19,CARD_1:79 .= n - (m + 1) by XCMPLX_1:36; A25: C = Cw \/ {w} by A21,Lm1; A26: C \/ Bv \ {w} \/ {v} = (Cw \/ (Bv \ {w})) \/ {v} by XBOOLE_1:42 .= Cw \/ ((Bv \ {w}) \/ {v}) by XBOOLE_1:4; Bv \ {w} c= Bv by XBOOLE_1:36; then (Bv \ {w}) \/ {v} c= Bv \/ {v} by XBOOLE_1:9; then Cw \/ ((Bv \ {w}) \/ {v}) c= Cw \/ (Bv \/ {v}) by XBOOLE_1:9; then Cw \/ ((Bv \ {w}) \/ {v}) c= B \/ Cw by A11,Lm1; then Lin(C \/ Bv \ {w} \/ {v}) is Subspace of Lin(B \/ Cw) by A26,VECTSP_7: 18; then A27: w in Lin(B \/ Cw) by A22,VECTSP_4:16; now let x be set; assume x in Bv \/ C; then x in Bv or x in C by XBOOLE_0:def 2; then x in B or x in Cw or x in {w} by A13,A25,XBOOLE_0:def 2; then x in B \/ Cw or x in {w} by XBOOLE_0:def 2; then x in Lin(B \/ Cw) or x = w by TARSKI:def 1,VECTSP_7:13; hence x in the carrier of Lin(B \/ Cw) by A27,RLVECT_1:def 1; end; then Bv \/ C c= the carrier of Lin(B \/ Cw) by TARSKI:def 3; then Lin(Bv \/ C) is Subspace of Lin(B \/ Cw) by Th20; then the carrier of Lin(Bv \/ C) c= the carrier of Lin(B \/ Cw) & the carrier of Lin(B \/ Cw) c= the carrier of V by VECTSP_4:def 2; then the carrier of Lin(B \/ Cw) = the carrier of V by A20,XBOOLE_0:def 10; then the VectSpStr of V = Lin(B \/ Cw) by A20,VECTSP_4:37; hence thesis by A17,A23,A24,NAT_1:38; end; for m holds P[m] from Ind(A3, A5); hence thesis by A1,A2; end; begin :: :: Finite-Dimensional Vector Spaces :: definition let GF be Field, V be VectSp of GF; redefine attr V is finite-dimensional means :Def1: ex I being finite Subset of V st I is Basis of V; compatibility by MATRLIN:def 3; end; theorem Th24: V is finite-dimensional implies for I being Basis of V holds I is finite proof assume V is finite-dimensional; then consider A being finite Subset of V such that A1: A is Basis of V by Def1; let B be Basis of V; consider p being FinSequence such that A2: rng p = A by FINSEQ_1:73; reconsider p as FinSequence of the carrier of V by A2,FINSEQ_1:def 4; set Car = {Carrier(L) where L is Linear_Combination of B: ex i st i in dom p & Sum(L) = p.i}; set C = union Car; A3: C c= B proof let x be set; assume x in C; then consider R being set such that A4: x in R and A5: R in Car by TARSKI:def 4; consider L being Linear_Combination of B such that A6: R = Carrier(L) and ex i st i in dom p & Sum(L) = p.i by A5; R c= B by A6,VECTSP_6:def 7; hence x in B by A4; end; then C c= the carrier of V by XBOOLE_1:1; then reconsider C as Subset of V; A7: B is linearly-independent by VECTSP_7:def 3; then A8: C is linearly-independent by A3,VECTSP_7:2; for v being Vector of V holds v in (Omega).V iff v in Lin(C) proof let v be Vector of V; hereby assume v in (Omega).V; then v in the VectSpStr of V by VECTSP_4:def 4; then v in Lin(A) by A1,VECTSP_7:def 3; then consider LA being Linear_Combination of A such that A9: v = Sum(LA) by VECTSP_7:12; Carrier(LA) c= the carrier of Lin(C) proof let w be set; assume A10: w in Carrier(LA); then A11: w in Carrier(LA) & Carrier(LA) c= A by VECTSP_6:def 7; reconsider w'= w as Vector of V by A10; w' in Lin(B) by Th14; then consider LB being Linear_Combination of B such that A12: w = Sum(LB) by VECTSP_7:12; ex i being set st i in dom p & w = p.i by A2,A11,FUNCT_1:def 5; then A13: Carrier(LB) in Car by A12; Carrier(LB) c= C proof let x be set; assume x in Carrier(LB); hence x in C by A13,TARSKI:def 4; end; then LB is Linear_Combination of C by VECTSP_6:def 7; then w in Lin(C) by A12,VECTSP_7:12; hence w in the carrier of Lin(C) by RLVECT_1:def 1; end; then consider LC being Linear_Combination of C such that A14: Sum(LA) = Sum(LC) by Th10; thus v in Lin(C) by A9,A14,VECTSP_7:12; end; assume v in Lin(C); v in the carrier of the VectSpStr of V; then v in the carrier of (Omega).V by VECTSP_4:def 4; hence thesis by RLVECT_1:def 1; end; then (Omega).V = Lin(C) by VECTSP_4:38; then the VectSpStr of V = Lin(C) by VECTSP_4:def 4; then A15: C is Basis of V by A8,VECTSP_7:def 3; B c= C proof assume not B c= C; then consider v being set such that A16: v in B and A17: not v in C by TARSKI:def 3; set D = B \ C; A18: D misses C by XBOOLE_1:79; reconsider B as Subset of V; reconsider D as non empty Subset of V by A16,A17,XBOOLE_0:def 4; C \/ (B \ C) = C \/ B by XBOOLE_1:39 .= B by A3,XBOOLE_1:12; then B = C \/ D; hence contradiction by A7,A15,A18,Th19; end; then A19: B = C by A3,XBOOLE_0:def 10; defpred P[set, set] means ex L being Linear_Combination of B st $2 = Carrier(L) & Sum(L) = p.$1; A20: for i, y1, y2 st i in Seg len p & P[i, y1] & P[i, y2] holds y1 = y2 proof let i, y1, y2; assume that i in Seg len p and A21: P[i, y1] and A22: P[i, y2]; consider L1 being Linear_Combination of B such that A23: y1 = Carrier(L1) and A24: Sum(L1) = p.i by A21; consider L2 being Linear_Combination of B such that A25: y2 = Carrier(L2) and A26: Sum(L2) = p.i by A22; Carrier(L1) c= B & Carrier(L2) c= B by VECTSP_6:def 7; hence y1 = y2 by A7,A23,A24,A25,A26,MATRLIN:9; end; A27: for i st i in Seg len p ex x st P[i, x] proof let i; assume i in Seg len p; then i in dom p by FINSEQ_1:def 3; then p.i in the carrier of V by FINSEQ_2:13; then p.i in Lin(B) by Th14; then consider L being Linear_Combination of B such that A28: p.i = Sum(L) by VECTSP_7:12; P[i, Carrier(L)] by A28; hence thesis; end; ex q being FinSequence st dom q = Seg len p & for i st i in Seg len p holds P[i, q.i] from SeqEx(A20, A27); then consider q being FinSequence such that A29: dom q = Seg len p & for i st i in Seg len p holds P[i, q.i]; A30: dom p = dom q by A29,FINSEQ_1:def 3; now let x be set; assume x in rng q; then consider i being set such that A31: i in dom q and A32: x = q.i by FUNCT_1:def 5; reconsider i as Nat by A31; consider L being Linear_Combination of B such that A33: x = Carrier(L) and A34: Sum(L) = p.i by A29,A31,A32; thus x in Car by A30,A31,A33,A34; end; then A35: rng q c= Car by TARSKI:def 3; now let x be set; assume x in Car; then consider L being Linear_Combination of B such that A36: x = Carrier(L) and A37: ex i st i in dom p & Sum(L) = p.i; consider i such that A38: i in dom p & Sum(L) = p.i by A37; P[i, x] & P[i, q.i] by A29,A30,A36,A38; then x = q.i by A20,A29,A30,A38; hence x in rng q by A30,A38,FUNCT_1:def 5; end; then Car c= rng q by TARSKI:def 3; then A39: Car is finite by A35,XBOOLE_0:def 10; for R being set st R in Car holds R is finite proof let R be set; assume R in Car; then consider L being Linear_Combination of B such that A40: R = Carrier(L) and ex i st i in dom p & Sum(L) = p.i; thus R is finite by A40; end; hence B is finite by A19,A39,FINSET_1:25; end; theorem V is finite-dimensional implies for A being Subset of V st A is linearly-independent holds A is finite proof assume A1: V is finite-dimensional; let A be Subset of V; assume A is linearly-independent; then consider B being Basis of V such that A2: A c= B by VECTSP_7:27; B is finite by A1,Th24; hence A is finite by A2,FINSET_1:13; end; theorem Th26: V is finite-dimensional implies for A, B being Basis of V holds Card A = Card B proof assume A1: V is finite-dimensional; let A, B be Basis of V; reconsider A'= A, B'= B as finite Subset of V by A1,Th24; A2: the VectSpStr of V = Lin(A) by VECTSP_7:def 3; B' is linearly-independent by VECTSP_7:def 3; then A3: Card B' <= Card A' by A2,Th23; A4: the VectSpStr of V = Lin(B) by VECTSP_7:def 3; A' is linearly-independent by VECTSP_7:def 3; then Card A' <= Card B' by A4,Th23; hence Card A = Card B by A3,AXIOMS:21; end; theorem Th27: (0).V is finite-dimensional proof reconsider V'= (0).V as strict VectSp of GF; the carrier of V' = {0.V} by VECTSP_4:def 3 .= {0.V'} by VECTSP_4:19 .= the carrier of (0).V' by VECTSP_4:def 3; then A1: V' = (0).V' by VECTSP_4:39; reconsider I = {}(the carrier of V') as finite Subset of V' ; A2: I is linearly-independent by VECTSP_7:4; Lin(I) = (0).V' by VECTSP_7:14; then I is Basis of V' by A1,A2,VECTSP_7:def 3; hence thesis by Def1; end; theorem Th28: V is finite-dimensional implies W is finite-dimensional proof assume A1: V is finite-dimensional; consider A being Basis of W; consider I being Basis of V such that A2: A c= I by Th17; I is finite by A1,Th24; then A is finite by A2,FINSET_1:13; hence thesis by MATRLIN:def 3; end; definition let GF be Field, V be VectSp of GF; cluster strict finite-dimensional Subspace of V; existence proof take (0).V; thus thesis by Th27; end; end; definition let GF be Field, V be finite-dimensional VectSp of GF; cluster -> finite-dimensional Subspace of V; correctness by Th28; end; definition let GF be Field, V be finite-dimensional VectSp of GF; cluster strict Subspace of V; existence proof (0).V is strict finite-dimensional Subspace of V; hence thesis; end; end; begin :: :: Dimension of a Vector Space :: definition let GF be Field, V be VectSp of GF; assume A1: V is finite-dimensional; func dim V -> Nat means :Def2: for I being Basis of V holds it = Card I; existence proof consider A being finite Subset of V such that A2: A is Basis of V by A1,Def1; consider n being Nat such that A3: n = Card A; for I being Basis of V holds Card I = n by A1,A2,A3,Th26; hence thesis; end; uniqueness proof let n, m; assume that A4: for I being Basis of V holds Card I = n and A5: for I being Basis of V holds Card I = m; consider A being finite Subset of V such that A6: A is Basis of V by A1,Def1; Card A = n & Card A = m by A4,A5,A6; hence n = m; end; end; reserve V for finite-dimensional VectSp of GF, W, W1, W2 for Subspace of V, u, v for Vector of V; theorem Th29: dim W <= dim V proof reconsider V'= V as VectSp of GF; consider I being Basis of V'; A1: Lin(I) = the VectSpStr of V' by VECTSP_7:def 3; reconsider I as finite Subset of V by Th24; consider A being Basis of W; reconsider A as Subset of W; A is linearly-independent by VECTSP_7:def 3; then consider B being Subset of V such that A2: B is linearly-independent & B = A by Th15; reconsider A'= A as finite Subset of V by A2,Th24; A3: dim W = Card A by Def2; Card A' <= Card I by A1,A2,Th23; hence dim W <= dim V by A3,Def2; end; theorem Th30: for A being Subset of V st A is linearly-independent holds Card A = dim Lin(A) proof let A be Subset of V such that A1: A is linearly-independent; set W = Lin(A); now let x be set; assume x in A; then x in W by VECTSP_7:13; hence x in the carrier of W by RLVECT_1:def 1; end; then A c= the carrier of W by TARSKI:def 3; then consider B being Subset of W such that A2: B is linearly-independent & B = A by A1,Th16; W = Lin(B) by A2,Th21; then reconsider B as Basis of W by A2,VECTSP_7:def 3; Card B = dim W by Def2; hence Card A = dim Lin(A) by A2; end; theorem Th31: dim V = dim (Omega).V proof consider I being finite Subset of V such that A1: I is Basis of V by Def1; A2: Card I = dim V by A1,Def2; A3: I is linearly-independent by A1,VECTSP_7:def 3; (Omega).V = the VectSpStr of V by VECTSP_4:def 4 .= Lin(I) by A1,VECTSP_7:def 3; hence thesis by A2,A3,Th30; end; theorem dim V = dim W iff (Omega).V = (Omega).W proof hereby assume A1: dim V = dim W; consider A being Basis of W; consider B being Basis of V such that A2: A c= B by Th17; A3: Card A = dim V by A1,Def2 .= Card B by Def2; A c= the carrier of W & the carrier of W c= the carrier of V by VECTSP_4:def 2; then A c= the carrier of V & A is finite by Th24,XBOOLE_1:1; then reconsider A'= A as finite Subset of V; reconsider B'= B as finite Subset of V by Th24; A4: now assume A <> B; then A c< B by A2,XBOOLE_0:def 8; then Card A' < Card B' by CARD_2:67; hence contradiction by A3; end; reconsider A as Subset of W; reconsider B as Subset of V; (Omega).V = the VectSpStr of V by VECTSP_4:def 4 .= Lin(B) by VECTSP_7:def 3 .= Lin(A) by A4,Th21 .= the VectSpStr of W by VECTSP_7:def 3 .= (Omega).W by VECTSP_4:def 4; hence (Omega).V = (Omega).W; end; assume (Omega).V = (Omega).W; then A5: the VectSpStr of V = (Omega).W by VECTSP_4:def 4 .= the VectSpStr of W by VECTSP_4:def 4; consider A being finite Subset of V such that A6: A is Basis of V by Def1; consider B being finite Subset of W such that A7: B is Basis of W by Def1; A8: A is linearly-independent by A6,VECTSP_7:def 3; A9: B is linearly-independent by A7,VECTSP_7:def 3; A10: Lin(A) = the VectSpStr of W by A5,A6,VECTSP_7:def 3 .= Lin(B) by A7,VECTSP_7:def 3; reconsider A as Subset of V; reconsider B as Subset of W; dim V = Card A by A6,Def2 .= dim Lin(B) by A8,A10,Th30 .= Card B by A9,Th30 .= dim W by A7,Def2; hence dim V = dim W; end; theorem Th33: dim V = 0 iff (Omega).V = (0).V proof hereby assume A1: dim V = 0; consider I being finite Subset of V such that A2: I is Basis of V by Def1; Card I = 0 by A1,A2,Def2; then A3: I = {}(the carrier of V) by CARD_2:59; (Omega).V = the VectSpStr of V by VECTSP_4:def 4 .= Lin(I) by A2,VECTSP_7:def 3 .= (0).V by A3,VECTSP_7:14; hence (Omega).V = (0).V; end; assume (Omega).V = (0).V; then A4: the VectSpStr of V = (0).V by VECTSP_4:def 4; consider I being finite Subset of V such that A5: I is Basis of V by Def1; Lin(I) = (0).V by A4,A5,VECTSP_7:def 3; then A6: I = {} or I = {0.V} by VECTSP_7:15; now assume I = {0.V}; then I is linearly-dependent by VECTSP_7:5; hence contradiction by A5,VECTSP_7:def 3; end; hence dim V = 0 by A5,A6,Def2,CARD_1:47; end; theorem dim V = 1 iff ex v st v <> 0.V & (Omega).V = Lin{v} proof hereby consider I being finite Subset of V such that A1: I is Basis of V by Def1; assume dim V = 1; then Card I = 1 by A1,Def2; then consider v being set such that A2: I = {v} by CARD_2:60; v in I by A2,TARSKI:def 1; then reconsider v as Vector of V; {v} is linearly-independent & Lin{v} = the VectSpStr of V by A1,A2,VECTSP_7:def 3; then v <> 0.V & (Omega).V = Lin{v} by VECTSP_4:def 4,VECTSP_7:5; hence ex v st v <> 0.V & (Omega).V = Lin{v}; end; given v such that A3: v <> 0.V and A4: (Omega).V = Lin{v}; {v} is linearly-independent & Lin{v} = the VectSpStr of V by A3,A4,VECTSP_4:def 4,VECTSP_7:5; then {v} is Basis of V & Card {v} = 1 by CARD_1:79,VECTSP_7:def 3; hence dim V = 1 by Def2; end; theorem dim V = 2 iff ex u, v st u <> v & {u, v} is linearly-independent & (Omega).V = Lin{u, v} proof hereby consider I being finite Subset of V such that A1: I is Basis of V by Def1; assume dim V = 2; then A2: Card I = 2 by A1,Def2; then consider u being set such that A3: u in I by CARD_1:47,XBOOLE_0:def 1; reconsider u as Vector of V by A3; now assume I c= {u}; then card I <= card {u} by CARD_1:80; then 2 <= 1 by A2,CARD_1:79; hence contradiction; end; then consider v being set such that A4: v in I & not v in {u} by TARSKI:def 3; reconsider v as Vector of V by A4; A5: v <> u by A4,TARSKI:def 1; for x be set st x in {u, v} holds x in I by A3,A4,TARSKI:def 2; then A6: {u, v} c= I by TARSKI:def 3; now assume not I c= {u, v}; then consider w being set such that A7: w in I & not w in {u, v} by TARSKI:def 3; A8: w <> u & w <> v by A7,TARSKI:def 2; for x be set st x in {u, v, w} holds x in I by A3,A4,A7,ENUMSET1:13; then {u, v, w} c= I by TARSKI:def 3; then card {u, v, w} <= card I by CARD_1:80; then 3 <= 2 by A2,A5,A8,CARD_2:77; hence contradiction; end; then A9: I = {u, v} by A6,XBOOLE_0:def 10; then A10: Lin{u, v} = the VectSpStr of V by A1,VECTSP_7:def 3 .= (Omega).V by VECTSP_4:def 4; {u, v} is linearly-independent by A1,A9,VECTSP_7:def 3; hence ex u, v st u <> v & {u, v} is linearly-independent & (Omega).V = Lin{u, v} by A5,A10; end; given u, v such that A11: u <> v and A12: {u, v} is linearly-independent and A13: (Omega).V = Lin{u, v}; Lin{u, v} = the VectSpStr of V by A13,VECTSP_4:def 4; then {u, v} is Basis of V & Card {u, v} = 2 by A11,A12,CARD_2:76,VECTSP_7:def 3; hence dim V = 2 by Def2; end; theorem Th36: dim(W1 + W2) + dim(W1 /\ W2) = dim W1 + dim W2 proof reconsider V as VectSp of GF; reconsider W1, W2 as Subspace of V; A1: W1 /\ W2 is finite-dimensional by Th28; then consider I being finite Subset of W1 /\ W2 such that A2: I is Basis of W1 /\ W2 by Def1; A3: Card I = dim(W1 /\ W2) by A1,A2,Def2; W1 /\ W2 is Subspace of W1 by VECTSP_5:20; then consider I1 being Basis of W1 such that A4: I c= I1 by A2,Th17; reconsider I1 as finite Subset of W1 by Th24; W1 /\ W2 is Subspace of W2 by VECTSP_5:20; then consider I2 being Basis of W2 such that A5: I c= I2 by A2,Th17; reconsider I2 as finite Subset of W2 by Th24; A6: Card I2 = dim W2 by Def2; A7: W1 + W2 is finite-dimensional by Th28; set A = I1 \/ I2; now let v be set; assume v in A; then A8: v in I1 or v in I2 by XBOOLE_0:def 2; then A9: v in the carrier of W1 or v in the carrier of W2; the carrier of W1 c= the carrier of V & the carrier of W2 c= the carrier of V by VECTSP_4:def 2; then reconsider v'= v as Vector of V by A9; v' in W1 or v' in W2 by A8,RLVECT_1:def 1; then v' in W1 + W2 by VECTSP_5:6; hence v in the carrier of W1 + W2 by RLVECT_1:def 1; end; then A c= the carrier of W1 + W2 & A is finite by TARSKI:def 3; then reconsider A as finite Subset of W1 + W2; A10: I c= I1 /\ I2 by A4,A5,XBOOLE_1:19; now assume not I1 /\ I2 c= I; then consider x being set such that A11: x in I1 /\ I2 and A12: not x in I by TARSKI:def 3; x in I1 & x in I2 by A11,XBOOLE_0:def 3; then x in Lin(I1) & x in Lin(I2) by VECTSP_7:13; then x in the VectSpStr of W1 & x in the VectSpStr of W2 by VECTSP_7:def 3; then A13: x in the carrier of W1 & x in the carrier of W2 by RLVECT_1:def 1; then x in (the carrier of W1) /\ (the carrier of W2) by XBOOLE_0:def 3; then x in the carrier of W1 /\ W2 by VECTSP_5:def 2; then A14: x in the VectSpStr of W1 /\ W2 by RLVECT_1:def 1; A15: the carrier of W1 c= the carrier of V by VECTSP_4:def 2; then reconsider x'= x as Vector of V by A13; I c= the carrier of W1 /\ W2 & the carrier of W1 /\ W2 c= the carrier of V by VECTSP_4:def 2; then I c= the carrier of V by XBOOLE_1:1; then reconsider I'= I as Subset of V; now let y be set; assume y in I \/ {x}; then A16: y in I or y in {x} by XBOOLE_0:def 2; I c= the carrier of W1 /\ W2 & the carrier of W1 /\ W2 c= the carrier of V by VECTSP_4:def 2; then I c= the carrier of V by XBOOLE_1:1; then y in the carrier of V or y = x by A16,TARSKI:def 1; hence y in the carrier of V by A13,A15; end; then I \/ {x} c= the carrier of V by TARSKI:def 3; then reconsider Ix = I \/ {x} as Subset of V; I1 is linearly-independent & I1 is Subset of W1 by VECTSP_7:def 3; then consider I1' being Subset of V such that A17: I1' is linearly-independent & I1'= I1 by Th15; now let y be set; assume y in I \/ {x}; then y in I or y in {x} by XBOOLE_0:def 2; then y in I1 or y = x by A4,TARSKI:def 1; hence y in I1' by A11,A17,XBOOLE_0:def 3; end; then Ix c= I1' by TARSKI:def 3; then A18: Ix is linearly-independent by A17,VECTSP_7:2; x in {x} by TARSKI:def 1; then A19: x' in Ix by XBOOLE_0:def 2; Ix \ {x} = I \ {x} by XBOOLE_1:40 .= I by A12,ZFMISC_1:65; then A20: not x' in Lin(I') by A18,A19,Th18; Lin(I) = Lin(I') by Th21; hence contradiction by A2,A14,A20,VECTSP_7:def 3; end; then I = I1 /\ I2 by A10,XBOOLE_0:def 10; then A21: Card A = Card I1 + Card I2 - Card I by CARD_2:64; A c= the carrier of W1 + W2 & the carrier of W1 + W2 c= the carrier of V by VECTSP_4:def 2; then A c= the carrier of V by XBOOLE_1:1; then reconsider A'= A as Subset of V; A22: Lin(A') = Lin(A) by Th21; now let x be set; assume x in the carrier of W1 + W2; then x in W1 + W2 by RLVECT_1:def 1; then consider w1, w2 being Vector of V such that A23: w1 in W1 and A24: w2 in W2 and A25: x = w1 + w2 by VECTSP_5:5; reconsider w1 as Vector of W1 by A23,RLVECT_1:def 1; reconsider w2 as Vector of W2 by A24,RLVECT_1:def 1; w1 in Lin(I1) by Th14; then consider K1 being Linear_Combination of I1 such that A26: w1 = Sum(K1) by VECTSP_7:12; consider L1 being Linear_Combination of V such that A27: Carrier(L1) = Carrier(K1) & Sum(L1) = Sum(K1) by Th12; w2 in Lin(I2) by Th14; then consider K2 being Linear_Combination of I2 such that A28: w2 = Sum(K2) by VECTSP_7:12; consider L2 being Linear_Combination of V such that A29: Carrier(L2) = Carrier(K2) & Sum(L2) = Sum(K2) by Th12; set L = L1 + L2; Carrier(L1) c= I1 & Carrier(L2) c= I2 by A27,A29,VECTSP_6:def 7; then Carrier(L) c= Carrier(L1) \/ Carrier(L2) & Carrier(L1) \/ Carrier(L2) c= I1 \/ I2 by VECTSP_6:51,XBOOLE_1:13; then Carrier(L) c= I1 \/ I2 by XBOOLE_1:1; then reconsider L as Linear_Combination of A' by VECTSP_6:def 7; x = Sum(L) by A25,A26,A27,A28,A29,VECTSP_6:77; then x in Lin(A') by VECTSP_7:12; hence x in the carrier of Lin(A') by RLVECT_1:def 1; end; then the carrier of W1 + W2 c= the carrier of Lin(A') by TARSKI:def 3; then W1 + W2 is Subspace of Lin(A') by VECTSP_4:35; then A30: Lin(A) = W1 + W2 by A22,VECTSP_4:33; for L being Linear_Combination of A st Sum(L) = 0.(W1 + W2) holds Carrier(L) = {} proof let L be Linear_Combination of A; assume A31: Sum(L) = 0.(W1 + W2); A32: W1 is Subspace of W1 + W2 & W2 is Subspace of W1 + W2 by VECTSP_5:11; reconsider W1'= W1 as Subspace of W1 + W2 by VECTSP_5:11; reconsider W2'= W2 as Subspace of W1 + W2 by VECTSP_5:11; A33: Carrier(L) c= I1 \/ I2 by VECTSP_6:def 7; consider F being FinSequence of the carrier of W1 + W2 such that A34: F is one-to-one and A35: rng F = Carrier(L) and A36: Sum(L) = Sum(L (#) F) by VECTSP_6:def 9; set B = Carrier(L) /\ I1; reconsider B as Subset of rng F by A35,XBOOLE_1:17; consider P being Permutation of dom F such that A37: (F - B`) ^ (F - B) = F*P by A34,MATRLIN:8; reconsider F1 = F - B`, F2 = F - B as FinSequence of the carrier of W1 + W2 by FINSEQ_3:93; A38: F1 is one-to-one & F2 is one-to-one by A34,FINSEQ_3:94; consider L1 being Linear_Combination of W1 + W2 such that A39: Carrier(L1) = rng F1 /\ Carrier(L) and A40: L1 (#) F1 = L (#) F1 by Th8; Carrier(L1) c= rng F1 by A39,XBOOLE_1:17; then A41: Sum(L (#) F1) = Sum(L1) by A38,A40,Th7; rng F c= rng F; then reconsider X = rng F as Subset of rng F; X \ B` = X /\ B`` by SUBSET_1:32 .= B by XBOOLE_1:28; then rng F1 = B by FINSEQ_3:72; then A42: Carrier(L1) = I1 /\ (Carrier(L) /\ Carrier(L)) by A39,XBOOLE_1:16 .= Carrier(L) /\ I1; then A43: Carrier(L1) c= I1 & I1 c= the carrier of W1 by XBOOLE_1:17; then Carrier(L1) c= the carrier of W1' by XBOOLE_1:1; then consider K1 being Linear_Combination of W1' such that Carrier(K1) = Carrier(L1) and A44: Sum(K1) = Sum(L1) by Th13; consider L2 being Linear_Combination of W1 + W2 such that A45: Carrier(L2) = rng F2 /\ Carrier(L) and A46: L2 (#) F2 = L (#) F2 by Th8; Carrier(L2) c= rng F2 by A45,XBOOLE_1:17; then A47: Sum(L (#) F2) = Sum(L2) by A38,A46,Th7; A48: Carrier(L) \ I1 c= Carrier(L) by XBOOLE_1:36; rng F2 = Carrier(L) \ (Carrier(L) /\ I1) by A35,FINSEQ_3:72 .= Carrier(L) \ I1 by XBOOLE_1:47; then A49: Carrier(L2) = Carrier(L) \ I1 by A45,A48,XBOOLE_1:28; then Carrier(L2) c= I2 & I2 c= the carrier of W2 by A33,XBOOLE_1:43; then Carrier(L2) c= the carrier of W2' by XBOOLE_1:1; then consider K2 being Linear_Combination of W2' such that Carrier(K2) = Carrier(L2) and A50: Sum(K2) = Sum(L2) by Th13; A51: 0.(W1 + W2) = Sum(L (#) (F1^F2)) by A31,A36,A37,Th5 .= Sum((L (#) F1) ^ (L (#) F2)) by VECTSP_6:37 .= Sum(L1) + Sum(L2) by A41,A47,RLVECT_1:58; then Sum(L1) = - Sum(L2) by VECTSP_1:63 .= - Sum(K2) by A50,VECTSP_4:23; then Sum(K1) in W2 & Sum(K1) in W1 by A44,RLVECT_1:def 1; then Sum(K1) in W1 /\ W2 by VECTSP_5:7; then Sum(K1) in Lin(I) by A2,VECTSP_7:def 3; then consider KI being Linear_Combination of I such that A52: Sum(K1) = Sum(KI) by VECTSP_7:12; W1 /\ W2 is Subspace of W1 + W2 by VECTSP_5:29; then consider LI being Linear_Combination of W1 + W2 such that A53: Carrier(LI) = Carrier(KI) and A54: Sum(LI) = Sum(KI) by Th12; A55: Carrier(LI + L2) c= Carrier(LI) \/ Carrier(L2) by VECTSP_6:51; A56: I \/ I2 = I2 by A5,XBOOLE_1:12; Carrier(LI) c= I & Carrier(L2) c= I2 by A33,A49,A53,VECTSP_6:def 7, XBOOLE_1:43; then Carrier(LI) \/ Carrier(L2) c= I2 by A56,XBOOLE_1:13; then A57: Carrier(LI + L2) c= I2 & I2 c= the carrier of W2 by A55,XBOOLE_1:1; then Carrier(LI + L2) c= the carrier of W2 by XBOOLE_1:1; then consider K being Linear_Combination of W2 such that A58: Carrier(K) = Carrier(LI + L2) and A59: Sum(K) = Sum(LI + L2) by A32,Th13; reconsider K as Linear_Combination of I2 by A57,A58,VECTSP_6:def 7; I1 is Subset of W1 & I1 is linearly-independent by VECTSP_7:def 3; then consider I1' being Subset of W1 + W2 such that A60: I1' is linearly-independent & I1'= I1 by A32,Th15; Carrier(LI) c= I by A53,VECTSP_6:def 7; then Carrier(LI) c= I1' by A4,A60,XBOOLE_1:1; then A61: LI = L1 by A43,A44,A52,A54,A60,MATRLIN:9; A62: I2 is linearly-independent by VECTSP_7:def 3; 0.W2 = Sum(LI) + Sum(L2) by A44,A51,A52,A54,VECTSP_4:20 .= Sum(K) by A59,VECTSP_6:77; then A63: {} = Carrier(L1 + L2) by A58,A61,A62,VECTSP_7:def 1; A64: Carrier(L) = Carrier(L1) \/ Carrier(L2) by A42,A49,XBOOLE_1:51; A65: I1 misses (Carrier(L) \ I1) by XBOOLE_1:79; Carrier(L1) /\ Carrier(L2) = Carrier(L) /\ (I1 /\ (Carrier(L) \ I1)) by A42,A49,XBOOLE_1:16 .= Carrier(L) /\ {} by A65,XBOOLE_0:def 7 .= {}; then A66: Carrier(L1) misses Carrier(L2) by XBOOLE_0:def 7; now assume not Carrier(L) c= Carrier(L1 + L2); then consider x being set such that A67: x in Carrier(L) and A68: not x in Carrier(L1 + L2) by TARSKI:def 3; reconsider x as Vector of W1 + W2 by A67; A69: 0.GF = (L1 + L2).x by A68,VECTSP_6:20 .= L1.x + L2.x by VECTSP_6:def 11; per cases by A64,A67,XBOOLE_0:def 2; suppose A70: x in Carrier(L1); then consider v being Vector of W1 + W2 such that A71: x = v & L1.v <> 0.GF by VECTSP_6:19; not x in Carrier(L2) by A66,A70,XBOOLE_0:3; then L2.x = 0.GF by VECTSP_6:20; hence contradiction by A69,A71,RLVECT_1:10; suppose A72: x in Carrier(L2); then consider v being Vector of W1 + W2 such that A73: x = v & L2.v <> 0.GF by VECTSP_6:19; not x in Carrier(L1) by A66,A72,XBOOLE_0:3; then L1.x = 0.GF by VECTSP_6:20; hence contradiction by A69,A73,RLVECT_1:10; end; hence Carrier(L) = {} by A63,XBOOLE_1:3; end; then A is linearly-independent by VECTSP_7:def 1; then A is Basis of W1 + W2 by A30,VECTSP_7:def 3; then Card A = dim(W1 + W2) by A7,Def2; then dim(W1 + W2) + dim(W1 /\ W2) = Card I1 + Card I2 + - Card I + Card I by A3,A21,XCMPLX_0: def 8 .= Card I1 + Card I2 + (- Card I + Card I) by XCMPLX_1:1 .= Card I1 + Card I2 + 0 by XCMPLX_0:def 6 .= dim W1 + dim W2 by A6,Def2; hence thesis; end; theorem dim(W1 /\ W2) >= dim W1 + dim W2 - dim V proof A1: dim W1 + dim W2 - dim V = dim(W1 + W2) + dim(W1 /\ W2) - dim V by Th36 .= dim(W1 + W2) + (dim(W1 /\ W2) - dim V) by XCMPLX_1:29 ; A2: dim(W1 + W2) <= dim V by Th29; dim V + (dim(W1 /\ W2) - dim V) = dim V + (dim(W1 /\ W2) + -dim V) by XCMPLX_0:def 8 .= dim(W1 /\ W2) + (dim V + -dim V) by XCMPLX_1:1 .= dim(W1 /\ W2) + 0 by XCMPLX_0:def 6 .= dim(W1 /\ W2); hence thesis by A1,A2,AXIOMS:24; end; theorem V is_the_direct_sum_of W1, W2 implies dim V = dim W1 + dim W2 proof assume V is_the_direct_sum_of W1, W2; then A1: the VectSpStr of V = W1 + W2 & W1 /\ W2 = (0).V by VECTSP_5:def 4; then (Omega).(W1 /\ W2) = (0).V by VECTSP_4:def 4 .= (0).(W1 /\ W2) by VECTSP_4:47; then dim(W1 /\ W2) = 0 by Th33; then dim W1 + dim W2 = dim(W1 + W2) + 0 by Th36 .= dim (Omega).V by A1,VECTSP_4:def 4 .= dim V by Th31; hence dim V = dim W1 + dim W2; end; :: :: Fixed-Dimensional Subspace Family and Pencil of Subspaces :: Lm2: n <= dim V implies ex W being strict Subspace of V st dim W = n proof assume A1: n <= dim V; consider I being finite Subset of V such that A2: I is Basis of V by Def1; n <= Card I by A1,A2,Def2; then consider A being finite Subset of I such that A3: Card A = n by Th1; A c= the carrier of V by XBOOLE_1:1; then reconsider A as Subset of V; reconsider W = Lin(A) as strict finite-dimensional Subspace of V; I is linearly-independent by A2,VECTSP_7:def 3; then A is linearly-independent by VECTSP_7:2; then dim W = n by A3,Th30; hence thesis; end; theorem n <= dim V iff ex W being strict Subspace of V st dim W = n by Lm2,Th29; definition let GF be Field, V be finite-dimensional VectSp of GF, n be Nat; func n Subspaces_of V -> set means :Def3: x in it iff ex W being strict Subspace of V st W = x & dim W = n; existence proof set S = {Lin(A) where A is Subset of V: A is linearly-independent & Card A = n}; A1: x in S iff ex W being strict Subspace of V st W = x & dim W = n proof hereby assume x in S; then consider A being Subset of V such that A2: x = Lin(A) and A3: A is linearly-independent and A4: Card A = n; reconsider W = x as strict Subspace of V by A2; dim W = n by A2,A3,A4,Th30; hence ex W being strict Subspace of V st W = x & dim W = n; end; given W being strict Subspace of V such that A5: W = x and A6: dim W = n; consider A being finite Subset of W such that A7: A is Basis of W by Def1; reconsider A as Subset of W; A is linearly-independent by A7,VECTSP_7:def 3; then consider B being Subset of V such that A8: B is linearly-independent and A9: B = A by Th15; A10: x = Lin(A) by A5,A7,VECTSP_7:def 3 .= Lin(B) by A9,Th21; then Card B = n by A5,A6,A8,Th30; hence x in S by A8,A10; end; take S; thus thesis by A1; end; uniqueness proof defpred P[set] means ex W being strict Subspace of V st W = $1 & dim W = n; thus for X1,X2 being set st (for x being set holds x in X1 iff P[x]) & (for x being set holds x in X2 iff P[x]) holds X1 = X2 from SetEq; end; end; theorem n <= dim V implies n Subspaces_of V is non empty proof assume n <= dim V; then consider W being strict Subspace of V such that A1: dim W = n by Lm2; thus n Subspaces_of V is non empty by A1,Def3; end; theorem dim V < n implies n Subspaces_of V = {} proof assume that A1: dim V < n and A2: n Subspaces_of V <> {}; consider x being set such that A3: x in n Subspaces_of V by A2,XBOOLE_0:def 1; consider W being strict Subspace of V such that W = x and A4: dim W = n by A3,Def3; thus contradiction by A1,A4,Th29; end; theorem n Subspaces_of W c= n Subspaces_of V proof let x be set; assume x in n Subspaces_of W; then consider W1 being strict Subspace of W such that A1: W1 = x and A2: dim W1 = n by Def3; reconsider W1 as strict Subspace of V by VECTSP_4:34; W1 in n Subspaces_of V by A2,Def3; hence x in n Subspaces_of V by A1; end;